In this article, we prove some fixed point theorems of Geraghty-type concerning the existence and uniqueness of fixed points under the setting of modular metric spaces. Also, we give an application of our main results to establish the existence and uniqueness of a solution to a nonhomogeneous linear parabolic partial differential equation in the last section.

Throughout this article, let ℝ+ denote the set of all positive real numbers and let ℝ+ denote the set of all nonnegative real numbers.

Since the year 1922, Banach's contraction principle, due to its simplicity and applicability, has became a very popular tool in modern analysis, especially in nonlinear analysis including its applications to differential and integral equations, variational inequality theory, complementarity problems, equilibrium problems, minimization problems and many others. Also, many authors have improved, extended and generalized this contraction principle in several ways (see e.g. [1–10]).

In 1973, Geraghty [11] gave an interesting generalization of the contraction principle using the class S of the functions β: ℝ+→ [0, 1) satisfying the following condition:

β(tn)→1implies tn→0.

Theorem 1.1. [11]Let (X, d) be a complete metric space and f be a self-mapping on X such that there existsβ∈Ssatisfying

d(fx,fy)≤β(d(x,y))d(x,y)

(1.1)

for all x, y∈X. Then the sequence {xn } defined by xn = fxn- 1for each n ≥ 1 converges to the unique fixed point of f in X.

Theorem 1.2. [12]Let (X, ⊑) be a partially ordered metric set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let f be a nondecreasing self-mapping on X which satisfies the inequality (1.1) whenever x, y∈X are comparable. Assume that f is either continuous or

ifanondecreasingsequencexnconvergestox*,thenxn⊑x*foreachn≥1.

(1.2)

If, additionally, the following condition is satisfied:

foranyx,y∈X,thereexistsz∈Xwhichiscomparabletobothxandy,

(1.3)

then the sequence {xn } converges to the unique fixed point of f in X.

Let Ψ denote the class of functions ψ: ℝ+→ℝ+ satisfying the following conditions:

Theorem 1.3. [13]Let (X, ⊑) be a partially ordered metric set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let f be a nondecreasing self-mapping on X such that there exists x0∈X with x0⊑fx0. Suppose that there existβ∈Sand ψ∈ Ψ such that

ψ(d(fx,fy))≤β(ψ(d(x,y)))ψ(d(x,y)),

whenever x, y∈X are comparable. Assume also that the condition (1.2) holds. Then f has a fixed point.

On the other hand, in 2010, Chistyakov [14] introduced the notion of a modular metric space which is raised in an attempt to avoid some restrictions of the concept of a modular space (for the literature of a modular space, see e.g. [15–21] and references therein). Some of the early investigations on metric fixed point theory in this space refer to [22–24].

For the rest of this section, we present some notions and basic facts of modular metric spaces.

Definition 1.4. [14] Let X be a nonempty set. A function ω: ℝ+ × X × X →ℝ+∪ {∞} is said to be a metric modular on X if, for all x, y, z∈X, the following conditions hold:

(a)

ωλ (x, y) = 0 for all λ > 0 if and only if x = y.

(b)

ωλ (x, y) = ωλ (y, x) for all λ > 0.

(c)

ωλ+μ(x, y) ≤ ωλ (x, z) + ωμ (z, y) for all λ, μ > 0.

For any xι∈X, the set Xω (xι ) = {x∈X: limλ→∞ωλ(x, xι ) = 0} is called a modular metric space generated by xι and induced by ω. If its generator xι does not play any role in the situation (that is, Xω is independent of generators), we write Xω instead of Xω (xι ).

Observe that a metric modular ω on X is nonincreasing with respect to λ > 0. We can simply show this assertion using the condition (c). For any x, y∈X and 0 < μ < λ, we have

ωλ(x,y)≤ωλ-μ(x,x)+ωμ(x,y)=ωμ(x,y).

(1.4)

For any x, y∈X and λ > 0, we set

ωλ+(x,y):=limϵ↓0ωλ+ϵ(x,y),ωλ-(x,y):=limϵ↓0ωλ-ϵ(x,y).

Consequently, from (1.4), it follows that

ωλ+(x,y)≤ωλ(x,y)≤ωλ-(x,y).

For any x, y∈X, if a metric modular ω on X possesses a finite value and ωλ (x, y) = ωμ (x, y) for all λ, μ > 0, then d(x, y): = ωλ (x, y) is a metric on X.

Later, Chaipunya et al. [23] has altered the notion of convergent and Cauchy sequences in modular metric spaces under the direction of Mongkolkeha et al. [24].

Definition 1.5. [23, 24] Let Xω be a modular metric space and {xn } be a sequence in Xω .

(1)

A point x∈Xω is called a limit of {xn } if, for each λ, ϵ > 0, there exists n0∈ℕ such that ωλ (xn , x) < ϵ for all n ≥ n0. A sequence that has a limit is said to be convergent (or converges to x), which is written as lim n→∞xn = x.

(2)

A sequence {xn } in Xω is said to be a Cauchy sequence if, for each λ, ϵ > 0, there exists n0∈ℕ such that ωλ (xn , xm ) < ϵ for all m, n ≥ n0.

(3)

If every Cauchy sequences in X converges, X is said to be complete.

In this article, we prove a generalization of Geraghty's theorem which also improves the result of Eshagi Gordji et al. [13] under the influence of a modular metric space. An application to partial differential equation is also provided.

Actually, Geraghty's class S is equivalent to the class S1 when ∞ is not considered. It follows that, for each m∈ {1, 2, ..., n}, if (β1,β2,…,βm)∈Sm, then (β1,β2,…,βm,θ,θ,…,θ⏟n-mentries)∈Sn, where θ denotes the zero function. Also, note that, if (β,β,…,β)⏟nentries∈Sn, then we also have the following:

β(tk)→1nimplies tk→0.

Besides, if (β1,β2…,βn)∈Sn, then π((β1,β2…,βn))∈Sn, where π((β1, β2 ..., βn )) is a permutation of (β1, β2 ..., βn ). It is also important to know that, if (β1,β2…,βn)∈Sn, then (βn1,βn2,…,βnm)∈Sm for each m∈ {1, 2, ..., n}, where each βni is selected from {β1, β2, ..., βn } and βni≠βnj for all i, j∈ {1, 2, ..., m}.

whereψ∈Ψ̄and(α,β,γ)∈S3with α(t) + 2 max{supt≥ 0β(t), supt≥ 0γ(t)} < 1. Assume also that the condition (1.2) holds. If there exists x0∈Xωsuch that ωλ (x0, fx0) < ∞ for all λ > 0, then the following hold:

(1)

f has a fixed point x∞∈Xω .

(2)

The sequence {fnx0} converges to x∞ .

Proof. It is clear that the sequence {fnx0} is nondecreasing. Assume that, for each n ≥ 1, there exists λn> 0 such that ωλn(fnx0,fn+1x0)≠0. Otherwise, the proof is complete. For each n ≥ 1, if 0 < λ ≤ λn , then we also have ωλ (fnx0, fn+1x0) ≠ 0. Since fnx0⊑fn+1x0, for any 0 < λ ≤ λn , we have

which is a contradiction of our assumption. Therefore, lim n→∞ψ(ωλ (fnx0, fn+1x0)) = 0 and so, we have limn→∞ωλ(fnx0, fn+1x0) = 0. Moreover, we have limn→∞ωλ (fnx0, fn+1x0) = 0 for all λ > 0.

Next, we show that {fnx0} is a Cauchy sequence. Assume the contrary. So, there exists λ0, ϵ0> 0 for which we can define two subsequences {fmkx0} and {fnkx0} of the sequence {fnx0} such that, for any nk> mk> k, ωλ0(fmkx0,fnkx0)≥ϵ0, but ωλ0(fmkx0,fnk-1x0)<ϵ0. Now, since fmkx0⊑fnkx0, we observe that

Proof. By Theorem 2.1, we know that f has a fixed point x∞∈Xω . Assume that y∞∈Xω is also another fixed point of f. Thus, we can find w∈Xω with w⊑fw and comparable to both x∞ and y∞ . It follows that fnw is comparable with both x∞ and y∞ for each n∈ℕ. Observe that, for any λ > 0,

Therefore, {ψ(ωλ (fnw, x∞ ))} is nonincreasing and bounded below. So, it converges to some real number h ≥ 0. Assume that h > 0. According to the proof of Theorem 2.1, we know that limn→∞ωλ(fnw, fn+1w) = 0 for all λ > 0. Thus, letting n → ∞ in the inequality (2.3), we have

1≤lim infn→∞ α(ψ(ωλ(fnw,x∞))).

Thus, we have {fnw} converges to x∞ . Similarly, we obtain that {fnw } converges also to y∞ . Since the limit is unique, we have x∞ = y∞ . This contradicts our assumption. Therefore, the theorem is proved. ■

Corollary 2.3. Additional to Theorem 2.1, if Xωis totally ordered, then the fixed point in Theorem 2.1 is unique.

Proof. Since Xω is totally ordered, the condition (2.2) is satisfied. Thus, applying Theorem 2.2, we obtain the result. ■

The following two corollaries nicely broaden the results in [24] (see Theorems 3.2 and 3.6 [24]).

Corollary 2.4. Let Xωbe a complete modular metric space with a partial ordering⊑and f be a self-mapping on Xωsuch that, for any λ > 0, there exists η(λ) ∈ (0, λ) such that

ψ(ωλ(fx,fy))≤α(ψ(ωλ(x,y)))ψ(ωλ+η(λ)(x,y)),

whereα∈Sandψ∈Ψ̄. Assume also that f is continuous or the condition (1.2) holds. Then f has a fixed point in Xω. Moreover, if the condition (2.2) is satisfied, the fixed point is unique.

Proof. Since α∈S, we have (α,θ,θ)∈S3. Thus, apply Theorems 2.1 and 2.2, we have the conclusion. ■

Corollary 2.5. Let Xωbe a complete modular metric space with a partial ordering⊑and f be a self-mapping on Xωsuch that, for any λ > 0, there exist ζ(λ), μ(λ) ∈ (0, λ) such that

ψ(ωλ(fx,fy))≤β(ψ(ωλ(x,y)))ψ(ωλ(x,fx))+γ(ψ(ωλ(x,y)))ψ(ωλ(y,fy)),

whereψ∈Ψ̄and(β,γ)∈S2with max{supt≥ 0β(t), supt≥ 0γ(t)} < 1. Assume also that f is continuous or that the condition (1.2) holds. Then f has a fixed point in Xω. Moreover, if the condition (2.2) is satisfied, the fixed point is unique.

Proof. Since (β,γ)∈S2, we have (θ,β,γ)∈S3. Thus, apply Theorems 2.1 and 2.2, we have the conclusion. ■

In this section, we give an application of our theorems to establish the existence and uniqueness of a solution to a nonhomogeneous linear parabolic partial differential equation satisfying a given initial condition.

Then, the existence and uniqueness of the solution of the system (3.1) is affirmative.

It is essential to note that the problem (3.1) is equivalent (under the assumption of Theorem 3.1) to the integral equation:

u(x,t)=∫-∞∞k(x-ξ,t)φ(ξ)dξ+∫0t∫-∞∞k(x-ξ,t-τ)F(ξ,τ,u(ξ,τ),ux(ξ,τ))dξdτ

(3.2)

for all x∈ℝ and 0 < t ≤ T, where

k(x,t):=14πte-x24t

for all x∈ℝ and t > 0. The system (3.1) possesses a unique solution if and only if the equation (3.2) possesses a unique solution u such that u and ux are both continuous and bounded for all x∈ℝ and 0 < t ≤ T.

For the case u = v, it is obvious that the above inequality is satisfied. Thus, we now have the inequality (2.1) holds for any comparable u, v∈ Ωω.

Note that any constant functions are contained in Ω. Now, for any u, v∈ Ωω, we may choose a constant function w∈ Ω ω for which u, v⊑w and θ⊑w. Consequently, we have w(x, t) = ||w|| for all (x, t) ∈ℝ× I. Also, observe that this w attains the following:

Acknowledgements

The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No.55000613). This research was partially finished at Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, Korea, while the first and third authors visit here. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (Grant no. 2011-0021821). Furthermore, the authors are grateful for the reviewers for the careful reading of the article and for the suggestions which improved the quality of this work.

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